How can I make a sliderule?

First, I'll note this interesting thread: #4158384.
However I unfortunately don't have privileges to post there so I'll start this new one.

I'm trying to make my own sliderule. How did they do it? How did Napier invent his "bones?" A more particular question would be: If I make a table of powers of 2, arraying them along a numberline, and a similar table of powers of 3 (9,27,81, etc) then, could I combine these? I know the answer is "yes" since that's what a sliderule DOES, but I don't know how to reconcile the two. Where, on the table of x2 logarithms, do I etch in the 3?

How would you do it from first principles? Assume you can only add and multiply: I can definitely type log(2) into my venerable HP15c simulator and the the answer but what if I was trying to INVENT logs, instead of using them to define themselves? That seems like cheating: circular.

Staff: Mentor

Then you would use the series for logarithms input 1, 2, 3... and use the output scaled to measure lengths on the rule.

However since we have this advanced technology called a calculator we can get the values that way. However if you sent back in time and had to construct one from memory then you'd need to use the Taylor series for the natural log.

Hey thanks! So I came up with this:
1) Make a slide rule based on base 2 as follows. .25, .5, 1,2,4,8,16...
2) Question is where does the 3 go? Well, we know other powers of 2. For instance sqrt(2) is just half the distance between the marks on the base 2 rule.
3) So find some sum of distances whose powers, 2^(1 + a/2 + b/4 + c/8) multiply to get 3. Not all the a,b,c have to be 1: some are zero.

explains how Napier calculated his logarithms. Given the calculation tools of Napier's time, parchment and quill pens, this was a lengthy process, which reportedly took Napier some 20 years to complete.

It was Henry Briggs who seized on Napier's ideas and produced the first table of logarithms. He also made logs more convenient to use, by switching from Napier's natural logarithms to the so-called common logs based on the number 10:

Napier did all of his natural log calculations without knowledge of the base of the natural logs, e ( ≈ 2.71828 ...), which is now called Euler's constant, but it was in fact discovered by Jacob Bernoulli many years before Euler was born: